예제 #1
0
int
mpfr_setsign (mpfr_ptr z, mpfr_srcptr x, int s, mp_rnd_t rnd_mode)
{
  return mpfr_set4 (z, x, rnd_mode, s ? -1 : 1);
}
예제 #2
0
파일: tanh.c 프로젝트: pgundlach/LuaTeX
int
mpfr_tanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
{
  /****** Declaration ******/
  mpfr_t x;
  int inexact;
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
     ("y[%Pu]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  /* Special value checking */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
    {
      if (MPFR_IS_NAN (xt))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (xt))
        {
          /* tanh(inf) = 1 && tanh(-inf) = -1 */
          return mpfr_set_si (y, MPFR_INT_SIGN (xt), rnd_mode);
        }
      else /* tanh (0) = 0 and xt is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO(xt));
          MPFR_SET_ZERO (y);
          MPFR_SET_SAME_SIGN (y, xt);
          MPFR_RET (0);
        }
    }

  /* tanh(x) = x - x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 0,
                                    rnd_mode, {});

  MPFR_TMP_INIT_ABS (x, xt);

  MPFR_SAVE_EXPO_MARK (expo);

  /* General case */
  {
    /* Declaration of the intermediary variable */
    mpfr_t t, te;
    mpfr_exp_t d;

    /* Declaration of the size variable */
    mpfr_prec_t Ny = MPFR_PREC(y);   /* target precision */
    mpfr_prec_t Nt;                  /* working precision */
    long int err;                  /* error */
    int sign = MPFR_SIGN (xt);
    MPFR_ZIV_DECL (loop);
    MPFR_GROUP_DECL (group);

    /* First check for BIG overflow of exp(2*x):
       For x > 0, exp(2*x) > 2^(2*x)
       If 2 ^(2*x) > 2^emax or x>emax/2, there is an overflow */
    if (MPFR_UNLIKELY (mpfr_cmp_si (x, __gmpfr_emax/2) >= 0)) {
      /* initialise of intermediary variables
         since 'set_one' label assumes the variables have been
         initialize */
      MPFR_GROUP_INIT_2 (group, MPFR_PREC_MIN, t, te);
      goto set_one;
    }

    /* Compute the precision of intermediary variable */
    /* The optimal number of bits: see algorithms.tex */
    Nt = Ny + MPFR_INT_CEIL_LOG2 (Ny) + 4;
    /* if x is small, there will be a cancellation in exp(2x)-1 */
    if (MPFR_GET_EXP (x) < 0)
      Nt += -MPFR_GET_EXP (x);

    /* initialise of intermediary variable */
    MPFR_GROUP_INIT_2 (group, Nt, t, te);

    MPFR_ZIV_INIT (loop, Nt);
    for (;;) {
      /* tanh = (exp(2x)-1)/(exp(2x)+1) */
      mpfr_mul_2ui (te, x, 1, MPFR_RNDN);  /* 2x */
      /* since x > 0, we can only have an overflow */
      mpfr_exp (te, te, MPFR_RNDN);        /* exp(2x) */
      if (MPFR_UNLIKELY (MPFR_IS_INF (te))) {
      set_one:
        inexact = MPFR_FROM_SIGN_TO_INT (sign);
        mpfr_set4 (y, __gmpfr_one, MPFR_RNDN, sign);
        if (MPFR_IS_LIKE_RNDZ (rnd_mode, MPFR_IS_NEG_SIGN (sign)))
          {
            inexact = -inexact;
            mpfr_nexttozero (y);
          }
        break;
      }
      d = MPFR_GET_EXP (te);              /* For Error calculation */
      mpfr_add_ui (t, te, 1, MPFR_RNDD);   /* exp(2x) + 1*/
      mpfr_sub_ui (te, te, 1, MPFR_RNDU);  /* exp(2x) - 1*/
      d = d - MPFR_GET_EXP (te);
      mpfr_div (t, te, t, MPFR_RNDN);      /* (exp(2x)-1)/(exp(2x)+1)*/

      /* Calculation of the error */
      d = MAX(3, d + 1);
      err = Nt - (d + 1);

      if (MPFR_LIKELY ((d <= Nt / 2) && MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
        {
          inexact = mpfr_set4 (y, t, rnd_mode, sign);
          break;
        }

      /* if t=1, we still can round since |sinh(x)| < 1 */
      if (MPFR_GET_EXP (t) == 1)
        goto set_one;

      /* Actualisation of the precision */
      MPFR_ZIV_NEXT (loop, Nt);
      MPFR_GROUP_REPREC_2 (group, Nt, t, te);
    }
    MPFR_ZIV_FREE (loop);
    MPFR_GROUP_CLEAR (group);
  }
  MPFR_SAVE_EXPO_FREE (expo);
  inexact = mpfr_check_range (y, inexact, rnd_mode);

  return inexact;
}
예제 #3
0
파일: sinh.c 프로젝트: Distrotech/mpfr
int
mpfr_sinh (mpfr_ptr y, mpfr_srcptr xt, mpfr_rnd_t rnd_mode)
{
  mpfr_t x;
  int inexact;

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
     ("y[%Pu]=%.*Rg inexact=%d",
      mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
    {
      if (MPFR_IS_NAN (xt))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else if (MPFR_IS_INF (xt))
        {
          MPFR_SET_INF (y);
          MPFR_SET_SAME_SIGN (y, xt);
          MPFR_RET (0);
        }
      else /* xt is zero */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (xt));
          MPFR_SET_ZERO (y);   /* sinh(0) = 0 */
          MPFR_SET_SAME_SIGN (y, xt);
          MPFR_RET (0);
        }
    }

  /* sinh(x) = x + x^3/6 + ... so the error is < 2^(3*EXP(x)-2) */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP(xt), 2, 1,
                                    rnd_mode, {});

  MPFR_TMP_INIT_ABS (x, xt);

  {
    mpfr_t t, ti;
    mpfr_exp_t d;
    mpfr_prec_t Nt;    /* Precision of the intermediary variable */
    long int err;    /* Precision of error */
    MPFR_ZIV_DECL (loop);
    MPFR_SAVE_EXPO_DECL (expo);
    MPFR_GROUP_DECL (group);

    MPFR_SAVE_EXPO_MARK (expo);

    /* compute the precision of intermediary variable */
    Nt = MAX (MPFR_PREC (x), MPFR_PREC (y));
    /* the optimal number of bits : see algorithms.ps */
    Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;
    /* If x is near 0, exp(x) - 1/exp(x) = 2*x+x^3/3+O(x^5) */
    if (MPFR_GET_EXP (x) < 0)
      Nt -= 2*MPFR_GET_EXP (x);

    /* initialise of intermediary variables */
    MPFR_GROUP_INIT_2 (group, Nt, t, ti);

    /* First computation of sinh */
    MPFR_ZIV_INIT (loop, Nt);
    for (;;)
      {
        MPFR_BLOCK_DECL (flags);

        /* compute sinh */
        MPFR_BLOCK (flags, mpfr_exp (t, x, MPFR_RNDD));
        if (MPFR_OVERFLOW (flags))
          /* exp(x) does overflow */
          {
            /* sinh(x) = 2 * sinh(x/2) * cosh(x/2) */
            mpfr_div_2ui (ti, x, 1, MPFR_RNDD); /* exact */

            /* t <- cosh(x/2): error(t) <= 1 ulp(t) */
            MPFR_BLOCK (flags, mpfr_cosh (t, ti, MPFR_RNDD));
            if (MPFR_OVERFLOW (flags))
              /* when x>1 we have |sinh(x)| >= cosh(x/2), so sinh(x)
                 overflows too */
              {
                inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
                break;
              }

            /* ti <- sinh(x/2): , error(ti) <= 1 ulp(ti)
               cannot overflow because 0 < sinh(x) < cosh(x) when x > 0 */
            mpfr_sinh (ti, ti, MPFR_RNDD);

            /* multiplication below, error(t) <= 5 ulp(t) */
            MPFR_BLOCK (flags, mpfr_mul (t, t, ti, MPFR_RNDD));
            if (MPFR_OVERFLOW (flags))
              {
                inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
                break;
              }

            /* doubling below, exact */
            MPFR_BLOCK (flags, mpfr_mul_2ui (t, t, 1, MPFR_RNDN));
            if (MPFR_OVERFLOW (flags))
              {
                inexact = mpfr_overflow (y, rnd_mode, MPFR_SIGN (xt));
                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
                break;
              }

            /* we have lost at most 3 bits of precision */
            err = Nt - 3;
            if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y),
                                             rnd_mode)))
              {
                inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
                break;
              }
            err = Nt; /* double the precision */
          }
        else
          {
            d = MPFR_GET_EXP (t);
            mpfr_ui_div (ti, 1, t, MPFR_RNDU); /* 1/exp(x) */
            mpfr_sub (t, t, ti, MPFR_RNDN);    /* exp(x) - 1/exp(x) */
            mpfr_div_2ui (t, t, 1, MPFR_RNDN);  /* 1/2(exp(x) - 1/exp(x)) */

            /* it may be that t is zero (in fact, it can only occur when te=1,
               and thus ti=1 too) */
            if (MPFR_IS_ZERO (t))
              err = Nt; /* double the precision */
            else
              {
                /* calculation of the error */
                d = d - MPFR_GET_EXP (t) + 2;
                /* error estimate: err = Nt-(__gmpfr_ceil_log2(1+pow(2,d)));*/
                err = Nt - (MAX (d, 0) + 1);
                if (MPFR_LIKELY (MPFR_CAN_ROUND (t, err, MPFR_PREC (y),
                                                 rnd_mode)))
                  {
                    inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));
                    break;
                  }
              }
          }

        /* actualisation of the precision */
        Nt += err;
        MPFR_ZIV_NEXT (loop, Nt);
        MPFR_GROUP_REPREC_2 (group, Nt, t, ti);
      }
    MPFR_ZIV_FREE (loop);
    MPFR_GROUP_CLEAR (group);
    MPFR_SAVE_EXPO_FREE (expo);
  }

  return mpfr_check_range (y, inexact, rnd_mode);
}
예제 #4
0
int
mpfr_atanh (mpfr_ptr y, mpfr_srcptr xt , mpfr_rnd_t rnd_mode)
{
  int inexact;
  mpfr_t x, t, te;
  mpfr_prec_t Nx, Ny, Nt;
  mpfr_exp_t err;
  MPFR_ZIV_DECL (loop);
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg rnd=%d", mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
    ("y[%Pu]=%.*Rg inexact=%d",
     mpfr_get_prec (y), mpfr_log_prec, y, inexact));

  /* Special cases */
  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
    {
      /* atanh(NaN) = NaN, and atanh(+/-Inf) = NaN since tanh gives a result
         between -1 and 1 */
      if (MPFR_IS_NAN (xt) || MPFR_IS_INF (xt))
        {
          MPFR_SET_NAN (y);
          MPFR_RET_NAN;
        }
      else /* necessarily xt is 0 */
        {
          MPFR_ASSERTD (MPFR_IS_ZERO (xt));
          MPFR_SET_ZERO (y);   /* atanh(0) = 0 */
          MPFR_SET_SAME_SIGN (y,xt);
          MPFR_RET (0);
        }
    }

  /* atanh (x) = NaN as soon as |x| > 1, and arctanh(+/-1) = +/-Inf */
  if (MPFR_UNLIKELY (MPFR_GET_EXP (xt) > 0))
    {
      if (MPFR_GET_EXP (xt) == 1 && mpfr_powerof2_raw (xt))
        {
          MPFR_SET_INF (y);
          MPFR_SET_SAME_SIGN (y, xt);
          mpfr_set_divby0 ();
          MPFR_RET (0);
        }
      MPFR_SET_NAN (y);
      MPFR_RET_NAN;
    }

  /* atanh(x) = x + x^3/3 + ... so the error is < 2^(3*EXP(x)-1) */
  MPFR_FAST_COMPUTE_IF_SMALL_INPUT (y, xt, -2 * MPFR_GET_EXP (xt), 1, 1,
                                    rnd_mode, {});

  MPFR_SAVE_EXPO_MARK (expo);

  /* Compute initial precision */
  Nx = MPFR_PREC (xt);
  MPFR_TMP_INIT_ABS (x, xt);
  Ny = MPFR_PREC (y);
  Nt = MAX (Nx, Ny);
  /* the optimal number of bits : see algorithms.ps */
  Nt = Nt + MPFR_INT_CEIL_LOG2 (Nt) + 4;

  /* initialise of intermediary variable */
  mpfr_init2 (t, Nt);
  mpfr_init2 (te, Nt);

  /* First computation of cosh */
  MPFR_ZIV_INIT (loop, Nt);
  for (;;)
    {
      /* compute atanh */
      mpfr_ui_sub (te, 1, x, MPFR_RNDU);   /* (1-xt)*/
      mpfr_add_ui (t,  x, 1, MPFR_RNDD);   /* (xt+1)*/
      mpfr_div (t, t, te, MPFR_RNDN);      /* (1+xt)/(1-xt)*/
      mpfr_log (t, t, MPFR_RNDN);          /* ln((1+xt)/(1-xt))*/
      mpfr_div_2ui (t, t, 1, MPFR_RNDN);   /* (1/2)*ln((1+xt)/(1-xt))*/

      /* error estimate: see algorithms.tex */
      /* FIXME: this does not correspond to the value in algorithms.tex!!! */
      /* err=Nt-__gmpfr_ceil_log2(1+5*pow(2,1-MPFR_EXP(t)));*/
      err = Nt - (MAX (4 - MPFR_GET_EXP (t), 0) + 1);

      if (MPFR_LIKELY (MPFR_IS_ZERO (t)
                       || MPFR_CAN_ROUND (t, err, Ny, rnd_mode)))
        break;

      /* reactualisation of the precision */
      MPFR_ZIV_NEXT (loop, Nt);
      mpfr_set_prec (t, Nt);
      mpfr_set_prec (te, Nt);
    }
  MPFR_ZIV_FREE (loop);

  inexact = mpfr_set4 (y, t, rnd_mode, MPFR_SIGN (xt));

  mpfr_clear(t);
  mpfr_clear(te);

  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (y, inexact, rnd_mode);
}
예제 #5
0
int
mpfr_copysign (mpfr_ptr z, mpfr_srcptr x, mpfr_srcptr y, mp_rnd_t rnd_mode)
{
  return mpfr_set4 (z, x, rnd_mode, MPFR_SIGN (y));
}
예제 #6
0
int
mpfr_sinh_cosh (mpfr_ptr sh, mpfr_ptr ch, mpfr_srcptr xt, mpfr_rnd_t rnd_mode)
{
    mpfr_t x;
    int inexact_sh, inexact_ch;

    MPFR_ASSERTN (sh != ch);

    MPFR_LOG_FUNC
    (("x[%Pu]=%.*Rg rnd=%d",
      mpfr_get_prec (xt), mpfr_log_prec, xt, rnd_mode),
     ("sh[%Pu]=%.*Rg ch[%Pu]=%.*Rg",
      mpfr_get_prec (sh), mpfr_log_prec, sh,
      mpfr_get_prec (ch), mpfr_log_prec, ch));

    if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (xt)))
    {
        if (MPFR_IS_NAN (xt))
        {
            MPFR_SET_NAN (ch);
            MPFR_SET_NAN (sh);
            MPFR_RET_NAN;
        }
        else if (MPFR_IS_INF (xt))
        {
            MPFR_SET_INF (sh);
            MPFR_SET_SAME_SIGN (sh, xt);
            MPFR_SET_INF (ch);
            MPFR_SET_POS (ch);
            MPFR_RET (0);
        }
        else /* xt is zero */
        {
            MPFR_ASSERTD (MPFR_IS_ZERO (xt));
            MPFR_SET_ZERO (sh);                   /* sinh(0) = 0 */
            MPFR_SET_SAME_SIGN (sh, xt);
            inexact_sh = 0;
            inexact_ch = mpfr_set_ui (ch, 1, rnd_mode); /* cosh(0) = 1 */
            return INEX(inexact_sh,inexact_ch);
        }
    }

    /* Warning: if we use MPFR_FAST_COMPUTE_IF_SMALL_INPUT here, make sure
       that the code also works in case of overlap (see sin_cos.c) */

    MPFR_TMP_INIT_ABS (x, xt);

    {
        mpfr_t s, c, ti;
        mpfr_exp_t d;
        mpfr_prec_t N;    /* Precision of the intermediary variables */
        long int err;    /* Precision of error */
        MPFR_ZIV_DECL (loop);
        MPFR_SAVE_EXPO_DECL (expo);
        MPFR_GROUP_DECL (group);

        MPFR_SAVE_EXPO_MARK (expo);

        /* compute the precision of intermediary variable */
        N = MPFR_PREC (ch);
        N = MAX (N, MPFR_PREC (sh));
        /* the optimal number of bits : see algorithms.ps */
        N = N + MPFR_INT_CEIL_LOG2 (N) + 4;

        /* initialise of intermediary variables */
        MPFR_GROUP_INIT_3 (group, N, s, c, ti);

        /* First computation of sinh_cosh */
        MPFR_ZIV_INIT (loop, N);
        for (;;)
        {
            MPFR_BLOCK_DECL (flags);

            /* compute sinh_cosh */
            MPFR_BLOCK (flags, mpfr_exp (s, x, MPFR_RNDD));
            if (MPFR_OVERFLOW (flags))
                /* exp(x) does overflow */
            {
                /* since cosh(x) >= exp(x), cosh(x) overflows too */
                inexact_ch = mpfr_overflow (ch, rnd_mode, MPFR_SIGN_POS);
                /* sinh(x) may be representable */
                inexact_sh = mpfr_sinh (sh, xt, rnd_mode);
                MPFR_SAVE_EXPO_UPDATE_FLAGS (expo, MPFR_FLAGS_OVERFLOW);
                break;
            }
            d = MPFR_GET_EXP (s);
            mpfr_ui_div (ti, 1, s, MPFR_RNDU);  /* 1/exp(x) */
            mpfr_add (c, s, ti, MPFR_RNDU);     /* exp(x) + 1/exp(x) */
            mpfr_sub (s, s, ti, MPFR_RNDN);     /* exp(x) - 1/exp(x) */
            mpfr_div_2ui (c, c, 1, MPFR_RNDN);  /* 1/2(exp(x) + 1/exp(x)) */
            mpfr_div_2ui (s, s, 1, MPFR_RNDN);  /* 1/2(exp(x) - 1/exp(x)) */

            /* it may be that s is zero (in fact, it can only occur when exp(x)=1,
               and thus ti=1 too) */
            if (MPFR_IS_ZERO (s))
                err = N; /* double the precision */
            else
            {
                /* calculation of the error */
                d = d - MPFR_GET_EXP (s) + 2;
                /* error estimate: err = N-(__gmpfr_ceil_log2(1+pow(2,d)));*/
                err = N - (MAX (d, 0) + 1);
                if (MPFR_LIKELY (MPFR_CAN_ROUND (s, err, MPFR_PREC (sh),
                                                 rnd_mode) &&               \
                                 MPFR_CAN_ROUND (c, err, MPFR_PREC (ch),
                                                 rnd_mode)))
                {
                    inexact_sh = mpfr_set4 (sh, s, rnd_mode, MPFR_SIGN (xt));
                    inexact_ch = mpfr_set (ch, c, rnd_mode);
                    break;
                }
            }
            /* actualisation of the precision */
            N += err;
            MPFR_ZIV_NEXT (loop, N);
            MPFR_GROUP_REPREC_3 (group, N, s, c, ti);
        }
        MPFR_ZIV_FREE (loop);
        MPFR_GROUP_CLEAR (group);
        MPFR_SAVE_EXPO_FREE (expo);
    }

    /* now, let's raise the flags if needed */
    inexact_sh = mpfr_check_range (sh, inexact_sh, rnd_mode);
    inexact_ch = mpfr_check_range (ch, inexact_ch, rnd_mode);

    return INEX(inexact_sh,inexact_ch);
}
예제 #7
0
파일: yn.c 프로젝트: Kirija/XPIR
int
mpfr_yn (mpfr_ptr res, long n, mpfr_srcptr z, mpfr_rnd_t r)
{
  int inex;
  unsigned long absn;
  MPFR_SAVE_EXPO_DECL (expo);

  MPFR_LOG_FUNC
    (("n=%ld x[%Pu]=%.*Rg rnd=%d", n, mpfr_get_prec (z), mpfr_log_prec, z, r),
     ("y[%Pu]=%.*Rg inexact=%d", mpfr_get_prec (res), mpfr_log_prec, res, inex));

  absn = SAFE_ABS (unsigned long, n);

  if (MPFR_UNLIKELY (MPFR_IS_SINGULAR (z)))
    {
      if (MPFR_IS_NAN (z))
        {
          MPFR_SET_NAN (res); /* y(n,NaN) = NaN */
          MPFR_RET_NAN;
        }
      /* y(n,z) tends to zero when z goes to +Inf, oscillating around
         0. We choose to return +0 in that case. */
      else if (MPFR_IS_INF (z))
        {
          if (MPFR_SIGN(z) > 0)
            return mpfr_set_ui (res, 0, r);
          else /* y(n,-Inf) = NaN */
            {
              MPFR_SET_NAN (res);
              MPFR_RET_NAN;
            }
        }
      else /* y(n,z) tends to -Inf for n >= 0 or n even, to +Inf otherwise,
              when z goes to zero */
        {
          MPFR_SET_INF(res);
          if (n >= 0 || ((unsigned long) n & 1) == 0)
            MPFR_SET_NEG(res);
          else
            MPFR_SET_POS(res);
          mpfr_set_divby0 ();
          MPFR_RET(0);
        }
    }

  /* for z < 0, y(n,z) is imaginary except when j(n,|z|) = 0, which we
     assume does not happen for a rational z. */
  if (MPFR_SIGN(z) < 0)
    {
      MPFR_SET_NAN (res);
      MPFR_RET_NAN;
    }

  /* now z is not singular, and z > 0 */

  MPFR_SAVE_EXPO_MARK (expo);

  /* Deal with tiny arguments. We have:
     y0(z) = 2 log(z)/Pi + 2 (euler - log(2))/Pi + O(log(z)*z^2), more
     precisely for 0 <= z <= 1/2, with g(z) = 2/Pi + 2(euler-log(2))/Pi/log(z),
                g(z) - 0.41*z^2 < y0(z)/log(z) < g(z)
     thus since log(z) is negative:
             g(z)*log(z) < y0(z) < (g(z) - z^2/2)*log(z)
     and since |g(z)| >= 0.63 for 0 <= z <= 1/2, the relative error on
     y0(z)/log(z) is bounded by 0.41*z^2/0.63 <= 0.66*z^2.
     Note: we use both the main term in log(z) and the constant term, because
     otherwise the relative error would be only in 1/log(|log(z)|).
  */
  if (n == 0 && MPFR_EXP(z) < - (mpfr_exp_t) (MPFR_PREC(res) / 2))
    {
      mpfr_t l, h, t, logz;
      mpfr_prec_t prec;
      int ok, inex2;

      prec = MPFR_PREC(res) + 10;
      mpfr_init2 (l, prec);
      mpfr_init2 (h, prec);
      mpfr_init2 (t, prec);
      mpfr_init2 (logz, prec);
      /* first enclose log(z) + euler - log(2) = log(z/2) + euler */
      mpfr_log (logz, z, MPFR_RNDD);    /* lower bound of log(z) */
      mpfr_set (h, logz, MPFR_RNDU);    /* exact */
      mpfr_nextabove (h);              /* upper bound of log(z) */
      mpfr_const_euler (t, MPFR_RNDD);  /* lower bound of euler */
      mpfr_add (l, logz, t, MPFR_RNDD); /* lower bound of log(z) + euler */
      mpfr_nextabove (t);              /* upper bound of euler */
      mpfr_add (h, h, t, MPFR_RNDU);    /* upper bound of log(z) + euler */
      mpfr_const_log2 (t, MPFR_RNDU);   /* upper bound of log(2) */
      mpfr_sub (l, l, t, MPFR_RNDD);    /* lower bound of log(z/2) + euler */
      mpfr_nextbelow (t);              /* lower bound of log(2) */
      mpfr_sub (h, h, t, MPFR_RNDU);    /* upper bound of log(z/2) + euler */
      mpfr_const_pi (t, MPFR_RNDU);     /* upper bound of Pi */
      mpfr_div (l, l, t, MPFR_RNDD);    /* lower bound of (log(z/2)+euler)/Pi */
      mpfr_nextbelow (t);              /* lower bound of Pi */
      mpfr_div (h, h, t, MPFR_RNDD);    /* upper bound of (log(z/2)+euler)/Pi */
      mpfr_mul_2ui (l, l, 1, MPFR_RNDD); /* lower bound on g(z)*log(z) */
      mpfr_mul_2ui (h, h, 1, MPFR_RNDU); /* upper bound on g(z)*log(z) */
      /* we now have l <= g(z)*log(z) <= h, and we need to add -z^2/2*log(z)
         to h */
      mpfr_mul (t, z, z, MPFR_RNDU);     /* upper bound on z^2 */
      /* since logz is negative, a lower bound corresponds to an upper bound
         for its absolute value */
      mpfr_neg (t, t, MPFR_RNDD);
      mpfr_div_2ui (t, t, 1, MPFR_RNDD);
      mpfr_mul (t, t, logz, MPFR_RNDU); /* upper bound on z^2/2*log(z) */
      mpfr_add (h, h, t, MPFR_RNDU);
      inex = mpfr_prec_round (l, MPFR_PREC(res), r);
      inex2 = mpfr_prec_round (h, MPFR_PREC(res), r);
      /* we need h=l and inex=inex2 */
      ok = (inex == inex2) && mpfr_equal_p (l, h);
      if (ok)
        mpfr_set (res, h, r); /* exact */
      mpfr_clear (l);
      mpfr_clear (h);
      mpfr_clear (t);
      mpfr_clear (logz);
      if (ok)
        goto end;
    }

  /* small argument check for y1(z) = -2/Pi/z + O(log(z)):
     for 0 <= z <= 1, |y1(z) + 2/Pi/z| <= 0.25 */
  if (n == 1 && MPFR_EXP(z) + 1 < - (mpfr_exp_t) MPFR_PREC(res))
    {
      mpfr_t y;
      mpfr_prec_t prec;
      mpfr_exp_t err1;
      int ok;
      MPFR_BLOCK_DECL (flags);

      /* since 2/Pi > 0.5, and |y1(z)| >= |2/Pi/z|, if z <= 2^(-emax-1),
         then |y1(z)| > 2^emax */
      prec = MPFR_PREC(res) + 10;
      mpfr_init2 (y, prec);
      mpfr_const_pi (y, MPFR_RNDU); /* Pi*(1+u)^2, where here and below u
                                      represents a quantity <= 1/2^prec */
      mpfr_mul (y, y, z, MPFR_RNDU); /* Pi*z * (1+u)^4, upper bound */
      MPFR_BLOCK (flags, mpfr_ui_div (y, 2, y, MPFR_RNDZ));
      /* 2/Pi/z * (1+u)^6, lower bound, with possible overflow */
      if (MPFR_OVERFLOW (flags))
        {
          mpfr_clear (y);
          MPFR_SAVE_EXPO_FREE (expo);
          return mpfr_overflow (res, r, -1);
        }
      mpfr_neg (y, y, MPFR_RNDN);
      /* (1+u)^6 can be written 1+7u [for another value of u], thus the
         error on 2/Pi/z is less than 7ulp(y). The truncation error is less
         than 1/4, thus if ulp(y)>=1/4, the total error is less than 8ulp(y),
         otherwise it is less than 1/4+7/8 <= 2. */
      if (MPFR_EXP(y) + 2 >= MPFR_PREC(y)) /* ulp(y) >= 1/4 */
        err1 = 3;
      else /* ulp(y) <= 1/8 */
        err1 = (mpfr_exp_t) MPFR_PREC(y) - MPFR_EXP(y) + 1;
      ok = MPFR_CAN_ROUND (y, prec - err1, MPFR_PREC(res), r);
      if (ok)
        inex = mpfr_set (res, y, r);
      mpfr_clear (y);
      if (ok)
        goto end;
    }

  /* we can use the asymptotic expansion as soon as z > p log(2)/2,
     but to get some margin we use it for z > p/2 */
  if (mpfr_cmp_ui (z, MPFR_PREC(res) / 2 + 3) > 0)
    {
      inex = mpfr_yn_asympt (res, n, z, r);
      if (inex != 0)
        goto end;
    }

  /* General case */
  {
    mpfr_prec_t prec;
    mpfr_exp_t err1, err2, err3;
    mpfr_t y, s1, s2, s3;
    MPFR_ZIV_DECL (loop);

    mpfr_init (y);
    mpfr_init (s1);
    mpfr_init (s2);
    mpfr_init (s3);

    prec = MPFR_PREC(res) + 2 * MPFR_INT_CEIL_LOG2 (MPFR_PREC (res)) + 13;
    MPFR_ZIV_INIT (loop, prec);
    for (;;)
      {
        mpfr_set_prec (y, prec);
        mpfr_set_prec (s1, prec);
        mpfr_set_prec (s2, prec);
        mpfr_set_prec (s3, prec);

        mpfr_mul (y, z, z, MPFR_RNDN);
        mpfr_div_2ui (y, y, 2, MPFR_RNDN); /* z^2/4 */

        /* store (z/2)^n temporarily in s2 */
        mpfr_pow_ui (s2, z, absn, MPFR_RNDN);
        mpfr_div_2si (s2, s2, absn, MPFR_RNDN);

        /* compute S1 * (z/2)^(-n) */
        if (n == 0)
          {
            mpfr_set_ui (s1, 0, MPFR_RNDN);
            err1 = 0;
          }
        else
          err1 = mpfr_yn_s1 (s1, y, absn - 1);
        mpfr_div (s1, s1, s2, MPFR_RNDN); /* (z/2)^(-n) * S1 */
        /* See algorithms.tex: the relative error on s1 is bounded by
           (3n+3)*2^(e+1-prec). */
        err1 = MPFR_INT_CEIL_LOG2 (3 * absn + 3) + err1 + 1;
        /* rel_err(s1) <= 2^(err1-prec), thus err(s1) <= 2^err1 ulps */

        /* compute (z/2)^n * S3 */
        mpfr_neg (y, y, MPFR_RNDN); /* -z^2/4 */
        err3 = mpfr_yn_s3 (s3, y, s2, absn); /* (z/2)^n * S3 */
        /* the error on s3 is bounded by 2^err3 ulps */

        /* add s1+s3 */
        err1 += MPFR_EXP(s1);
        mpfr_add (s1, s1, s3, MPFR_RNDN);
        /* the error is bounded by 1/2 + 2^err1*2^(- EXP(s1))
           + 2^err3*2^(EXP(s3) - EXP(s1)) */
        err3 += MPFR_EXP(s3);
        err1 = (err3 > err1) ? err3 + 1 : err1 + 1;
        err1 -= MPFR_EXP(s1);
        err1 = (err1 >= 0) ? err1 + 1 : 1;
        /* now the error on s1 is bounded by 2^err1*ulp(s1) */

        /* compute S2 */
        mpfr_div_2ui (s2, z, 1, MPFR_RNDN); /* z/2 */
        mpfr_log (s2, s2, MPFR_RNDN); /* log(z/2) */
        mpfr_const_euler (s3, MPFR_RNDN);
        err2 = MPFR_EXP(s2) > MPFR_EXP(s3) ? MPFR_EXP(s2) : MPFR_EXP(s3);
        mpfr_add (s2, s2, s3, MPFR_RNDN); /* log(z/2) + gamma */
        err2 -= MPFR_EXP(s2);
        mpfr_mul_2ui (s2, s2, 1, MPFR_RNDN); /* 2*(log(z/2) + gamma) */
        mpfr_jn (s3, absn, z, MPFR_RNDN); /* Jn(z) */
        mpfr_mul (s2, s2, s3, MPFR_RNDN); /* 2*(log(z/2) + gamma)*Jn(z) */
        err2 += 4; /* the error on s2 is bounded by 2^err2 ulps, see
                      algorithms.tex */

        /* add all three sums */
        err1 += MPFR_EXP(s1); /* the error on s1 is bounded by 2^err1 */
        err2 += MPFR_EXP(s2); /* the error on s2 is bounded by 2^err2 */
        mpfr_sub (s2, s2, s1, MPFR_RNDN); /* s2 - (s1+s3) */
        err2 = (err1 > err2) ? err1 + 1 : err2 + 1;
        err2 -= MPFR_EXP(s2);
        err2 = (err2 >= 0) ? err2 + 1 : 1;
        /* now the error on s2 is bounded by 2^err2*ulp(s2) */
        mpfr_const_pi (y, MPFR_RNDN); /* error bounded by 1 ulp */
        mpfr_div (s2, s2, y, MPFR_RNDN); /* error bounded by
                                           2^(err2+1)*ulp(s2) */
        err2 ++;

        if (MPFR_LIKELY (MPFR_CAN_ROUND (s2, prec - err2, MPFR_PREC(res), r)))
          break;
        MPFR_ZIV_NEXT (loop, prec);
      }
    MPFR_ZIV_FREE (loop);

    /* Assume two's complement for the test n & 1 */
    inex = mpfr_set4 (res, s2, r, n >= 0 || (n & 1) == 0 ?
                      MPFR_SIGN (s2) : - MPFR_SIGN (s2));

    mpfr_clear (y);
    mpfr_clear (s1);
    mpfr_clear (s2);
    mpfr_clear (s3);
  }

 end:
  MPFR_SAVE_EXPO_FREE (expo);
  return mpfr_check_range (res, inex, r);
}